The header <boost/operators.hpp> supplies
several sets of class templates (in namespace boost). These
templates define operators at namespace scope in terms of a minimal
number of fundamental operators provided by the class.

Overloaded operators for class types typically occur in groups. If you
can write x + y, you probably also want to be able
to write x += y. If you can write x < y, you
also want x > y, x >= y, and x <= y.
Moreover, unless your class has really surprising behavior, some of these
related operators can be defined in terms of others (e.g. x >= y
<=> !(x < y)). Replicating this boilerplate for multiple
classes is both tedious and error-prone. The boost/operators.hpp templates help
by generating operators for you at namespace scope based on other
operators you've defined in your class.

then the operators<>
template adds more than a dozen additional operators, such as
operator>, <=, >=, and
(binary) +. Two-argument forms of the
templates are also provided to allow interaction with other types.

Each operator template completes the concept(s) it describes by
defining overloaded operators for its target class.

The name of an operator class template indicates the concept that its target class will model.

Usually, the target class uses an instantation of the operator
class template as a base class. Some operator templates support an alternate method.

The concept can be compound, i.e. it may represent a common
combination of other, simpler concepts.

Most operator templates require their target class to support
operations related to the operators supplied by the template. In
accordance with widely accepted coding style recommendations, the
target class is often required to supply the assignment counterpart
operator of the concept's "main operator." For example, the
addable template requires operator+=(T
const&) and in turn supplies operator+(T const&, T
const&).

The discussed concepts are not necessarily the standard library's
concepts (CopyConstructible, etc.), although some of them could
be; they are what we call concepts with a small 'c'. In
particular, they are different from the former ones in that they do
not describe precise semantics of the operators they require to be
defined, except the requirements that (a) the semantics of the operators
grouped in one concept should be consistent (e.g. effects of
evaluating of a += b and
a = a + b expressions should be the
same), and (b) that the return types of the operators should follow
semantics of return types of corresponding operators for built-in types
(e.g.operator< should return a type convertible
to bool, and T::operator-= should return type
convertible to T). Such "loose" requirements make operators
library applicable to broader set of target classes from different
domains, i.e. eventually more useful.

The arguments to a binary operator commonly have identical types, but
it is not unusual to want to define operators which combine different
types. For example, one might want to multiply a
mathematical vector by a scalar. The two-argument template forms of the
arithmetic operator templates are supplied for this purpose. When
applying the two-argument form of a template, the desired return type of
the operators typically determines which of the two types in question
should be derived from the operator template. For example, if the result
of T + U is of type T, then
T (not U) should be derived from addable<T, U>. The comparison templates
(less_than_comparable<T,
U>, equality_comparable<T, U>,
equivalent<T, U>, and
partially_ordered<T,
U>) are exceptions to this guideline, since the return type
of the operators they define is bool.

On compilers which do not support partial specialization, the
two-argument forms must be specified by using the names shown below with
the trailing '2'. The single-argument forms with the
trailing '1' are provided for symmetry and to enable certain
applications of the base class chaining
technique.

Another application of the two-argument template forms is for mixed
arithmetics between a type T and a type U that
is convertible to T. In this case there are two ways where
the two-argument template forms are helpful: one is to provide the
respective signatures for operator overloading, the second is
performance.

With respect to the operator overloading assume e.g. that
U is int, that T is an
user-defined unlimited integer type, and that double
operator-(double, const T&) exists. If one wants to compute
int - T and does not provide T operator-(int, const
T&), the compiler will consider double operator-(double,
const T&) to be a better match than T operator-(const
T&, const T&), which will probably be different from the
user's intention. To define a complete set of operator signatures,
additional 'left' forms of the two-argument template forms are provided
(subtractable2_left<T,
U>, dividable2_left<T,
U>, modable2_left<T,
U>) that define the signatures for non-commutative
operators where U appears on the left hand side
(operator-(const U&, const T&),
operator/(const U&, const T&), operator%(const
U&, const T&)).

With respect to the performance observe that when one uses the single
type binary operator for mixed type arithmetics, the type U
argument has to be converted to type T. In practice,
however, there are often more efficient implementations of, say
T::operator-=(const U&) that avoid unnecessary
conversions from U to T. The two-argument
template forms of the arithmetic operator create additional operator
interfaces that use these more efficient implementations. There is,
however, no performance gain in the 'left' forms: they still need a
conversion from U to T and have an
implementation equivalent to the code that would be automatically created
by the compiler if it considered the single type binary operator to be
the best match.

Every operator class template, except the arithmetic examples and the iterator
helpers, has an additional, but optional, template type parameter
B. This parameter will be a publicly-derived base class of
the instantiated template. This means it must be a class type. It can be
used to avoid the bloating of object sizes that is commonly associated
with multiple-inheritance from several empty base classes (see the note for users of older versions for more
details). To provide support for a group of operators, use the
B parameter to chain operator templates into a single-base
class hierarchy, demostrated in the usage example.
The technique is also used by the composite operator templates to group
operator definitions. If a chain becomes too long for the compiler to
support, try replacing some of the operator templates with a single
grouped operator template that chains the old templates together; the
length limit only applies to the number of templates directly in the
chain, not those hidden in group templates.

Caveat: to chain to a base class which is
not a Boost operator template when using the single-argument form of a Boost operator template, you
must specify the operator template with the trailing '1' in
its name. Otherwise the library will assume you mean to define a binary
operation combining the class you intend to use as a base class and the
class you're deriving.

On some compilers (e.g. Borland, GCC) even single-inheritance
seems to cause an increase in object size in some cases. If you are not
defining a class template, you may get better object-size performance by
avoiding derivation altogether, and instead explicitly instantiating the
operator template as follows:

Note that some operator templates cannot use this workaround and must
be a base class of their primary operand type. Those templates define
operators which must be member functions, and the workaround needs the
operators to be independent friend functions. The relevant templates
are:

Any composite operator template that includes at least one of the
above

As Daniel Krügler pointed out, this technique violates 14.6.5/2
and is thus non-portable. The reasoning is, that the operators injected
by the instantiation of e.g.
less_than_comparable<myclass> can not be found
by ADL according to the rules given by 3.4.2/2, since myclass is
not an associated class of
less_than_comparable<myclass>.
Thus only use this technique if all else fails.

Many compilers (e.g. MSVC 6.3, GCC 2.95.2) will not enforce the
requirements in the operator template tables unless the operations which
depend on them are actually used. This is not standard-conforming
behavior. In particular, although it would be convenient to derive all
your classes which need binary operators from the operators<> and operators2<> templates, regardless of
whether they implement all the requirements of those templates, this
shortcut is not portable. Even if this currently works with your
compiler, it may not work later.

The arithmetic operator templates ease the task of creating a custom
numeric type. Given a core set of operators, the templates add related
operators to the numeric class. These operations are like the ones the
standard arithmetic types have, and may include comparisons, adding,
incrementing, logical and bitwise manipulations, etc. Further,
since most numeric types need more than one of these operators, some
templates are provided to combine several of the basic operator templates
in one declaration.

The requirements for the types used to instantiate the simple operator
templates are specified in terms of expressions which must be valid and
the expression's return type. The composite operator templates only list
what other templates they use. The supplied operations and requirements
of the composite operator templates can be inferred from the operations
and requirements of the listed components.

These templates are "simple" since they provide operators based on a
single operation the base type has to provide. They have an additional
optional template parameter B, which is not shown, for the
base class chaining technique.

The primary operand type T needs to be of class type,
built-in types are not supported.

Before talking about symmetry, we need to talk about optimizations to
understand the reasons for the different implementation styles of
operators. Let's have a look at operator+ for a class
T as an example:

T operator+( const T& lhs, const T& rhs )
{
return T( lhs ) += rhs;
}

This would be a normal implementation of operator+, but it
is not an efficient one. An unnamed local copy of lhs is
created, operator+= is called on it and it is copied to the
function return value (which is another unnamed object of type
T). The standard doesn't generally allow the intermediate
object to be optimized away:

3.7.2/2: Automatic storage duration

If a named automatic object has initialization or a destructor with
side effects, it shall not be destroyed before the end of its block,
nor shall it be eliminated as an optimization even if it appears to be
unused, except that a class object or its copy may be eliminated as
specified in 12.8.

The reference to 12.8 is important for us:

12.8/15: Copying class objects
...
For a function with a class return type, if the expression in the
return statement is the name of a local object, and the cv-unqualified
type of the local object is the same as the function return type, an
implementation is permitted to omit creating the temporary object to
hold the function return value, even if the class copy constructor or
destructor has side effects.

This optimization is known as the named return value optimization (NRVO),
which leads us to the following implementation for
operator+:

Given this implementation, the compiler is allowed to remove the
intermediate object. Sadly, not all compiler implement the NRVO, some
even implement it in an incorrect way which makes it useless here.
Without the NRVO, the NRVO-friendly code is no worse than the original
code showed above, but there is another possible implementation, which
has some very special properties:

T operator+( T lhs, const T& rhs )
{
return lhs += rhs;
}

The difference to the first implementation is that lhs is
not taken as a constant reference used to create a copy; instead,
lhs is a by-value parameter, thus it is already the copy
needed. This allows another optimization (12.2/2) for some cases.
Consider a + b + c where the result of
a + b is not copied when used as lhs
when adding c. This is more efficient than the original
code, but not as efficient as a compiler using the NRVO. For most people,
it is still preferable for compilers that don't implement the NRVO, but
the operator+ now has a different function signature. Also,
the number of objects created differs for
(a + b ) + c and
a + ( b + c ). Most probably,
this won't be a problem for you, but if your code relies on the function
signature or a strict symmetric behaviour, you should set
BOOST_FORCE_SYMMETRIC_OPERATORS in your user-config. This
will force the NRVO-friendly implementation to be used even for compilers
that don't implement the NRVO.

The following templates provide common groups of related operations.
For example, since a type which is addable is usually also subractable,
the additive template provides the
combined operators of both. The grouped operator templates have an
additional optional template parameter B, which is not
shown, for the base class chaining technique.

Spelling: euclidean vs. euclidian

Older versions of the Boost.Operators library used
"euclidian", but it was pointed out that
"euclidean" is the more common spelling.
To be compatible with older version, the library now supports
both spellings.

The arithmetic operator class templates operators<> and operators2<> are examples of
non-extensible operator grouping classes. These legacy class templates,
from previous versions of the header, cannot be used for base class chaining.

The operators_test.cpp
program demonstrates the use of the arithmetic operator templates, and
can also be used to verify correct operation. Check the compiler status
report for the test results with selected platforms.

The iterator helper templates ease the task of
creating a custom iterator. Similar to arithmetic types, a complete
iterator has many operators that are "redundant" and can be implemented
in terms of the core set of operators.

The dereference operators were motivated by
the iterator helpers, but are often useful in
non-iterator contexts as well. Many of the redundant iterator operators
are also arithmetic operators, so the iterator helper classes borrow many
of the operators defined above. In fact, only two new operators need to
be defined (the pointer-to-member operator-> and the
subscript operator[])!

The requirements for the types used to instantiate the dereference
operators are specified in terms of expressions which must be valid and
their return type. The composite operator templates list their component
templates, which the instantiating type must support, and possibly other
requirements.

There are five iterator operator class templates, each for a different
category of iterator. The following table shows the operator groups for
any category that a custom iterator could define. These class templates
have an additional optional template parameter B, which is
not shown, to support base class chaining.

There are also five iterator helper class templates, each
corresponding to a different iterator category. These classes cannot be
used for base class chaining. The following
summaries show that these class templates supply both the iterator
operators from the iterator operator class
templates and the iterator typedef's required by the C++ standard
(iterator_category, value_type,
etc.).

[1] Unlike other iterator helpers templates,
output_iterator_helper takes only one template parameter -
the type of its target class. Although to some it might seem like an
unnecessary restriction, the standard requires
difference_type and value_type of any output
iterator to be void (24.3.1 [lib.iterator.traits]), and
output_iterator_helper template respects this requirement.
Also, output iterators in the standard have void pointer and
reference types, so the output_iterator_helper
does the same.

[2] As self-proxying is the easiest and most common
way to implement output iterators (see, for example, insert [24.4.2] and
stream iterators [24.5] in the standard library),
output_iterator_helper supports the idiom by defining
operator* and operator++ member functions which
just return a non-const reference to the iterator itself. Support for
self-proxying allows us, in many cases, to reduce the task of writing an
output iterator to writing just two member functions - an appropriate
constructor and a copy-assignment operator. For example, here is a
possible implementation of boost::function_output_iterator
adaptor:

Note that support for self-proxying does not prevent you from using
output_iterator_helper to ease any other, different kind of
output iterator's implementation. If
output_iterator_helper's target type provides its own
definition of operator* or/and operator++, then
these operators will get used and the ones supplied by
output_iterator_helper will never be instantiated.

The iterators_test.cpp
program demonstrates the use of the iterator templates, and can also be
used to verify correct operation. The following is the custom iterator
defined in the test program. It demonstrates a correct (though trivial)
implementation of the core operations that must be defined in order for
the iterator helpers to "fill in" the rest of the iterator
operations.

The changes in the library interface and
recommended usage were motivated by some practical issues described
below. The new version of the library is still backward-compatible with
the former one (so you're not forced change any existing code),
but the old usage is deprecated. Though it was arguably simpler and more
intuitive than using base class chaining, it has
been discovered that the old practice of deriving from multiple operator
templates can cause the resulting classes to be much larger than they
should be. Most modern C++ compilers significantly bloat the size of
classes derived from multiple empty base classes, even though the base
classes themselves have no state. For instance, the size of
point<int> from the example
above was 12-24 bytes on various compilers for the Win32 platform,
instead of the expected 8 bytes.

Strictly speaking, it was not the library's fault--the language rules
allow the compiler to apply the empty base class optimization in that
situation. In principle an arbitrary number of empty base classes can be
allocated at the same offset, provided that none of them have a common
ancestor (see section 10.5 [class.derived] paragraph 5 of the standard).
But the language definition also doesn't require implementations
to do the optimization, and few if any of today's compilers implement it
when multiple inheritance is involved. What's worse, it is very unlikely
that implementors will adopt it as a future enhancement to existing
compilers, because it would break binary compatibility between code
generated by two different versions of the same compiler. As Matt Austern
said, "One of the few times when you have the freedom to do this sort of
thing is when you're targeting a new architecture...". On the other hand,
many common compilers will use the empty base optimization for single
inheritance hierarchies.

Given the importance of the issue for the users of the library (which
aims to be useful for writing light-weight classes like
MyInt or point<>), and the forces
described above, we decided to change the library interface so that the
object size bloat could be eliminated even on compilers that support only
the simplest form of the empty base class optimization. The current
library interface is the result of those changes. Though the new usage is
a bit more complicated than the old one, we think it's worth it to make
the library more useful in real world. Alexy Gurtovoy contributed the
code which supports the new usage idiom while allowing the library remain
backward-compatible.